Problem 45 - Fundamentals of Mathematics

Question

For the following problems, determine the missing numerator or denominator. $$\frac{4}{5}=\frac{?}{25}$$

Step-by-Step Solution

Answer

The missing numerator is 20.

Step 1: Identify the Relationship

First, observe how the two fractions are related. We can notice that the denominator of the second fraction is 25. Since the original fraction has a denominator of 5, determine how to transform 5 into 25.

Step 2: Find the Multiplier for the Denominator

To transform the denominator 5 into 25, you need to multiply by 5. This means multiplying both the numerator and denominator of the original fraction, \(\frac{4}{5}\), by 5 to keep the fractions equivalent.

Step 3: Apply the Multiplier to the Numerator

Since we determined to use the multiplier 5, apply it to the numerator as well: \(4 \times 5 = 20\).

Step 4: Write the Equivalent Fraction

Using these calculations, the equivalent fraction to \(\frac{4}{5}\) with a denominator of 25 is \(\frac{20}{25}\). The missing numerator is 20.

Key Concepts

Equivalent Fractions

In fractions, an equivalent fraction represents the same part of a whole, even if both the numerator and the denominator are different. This means that

• the ratios are equal
• the fractions have the same value, even though they look different

For example, \[\frac{1}{2} = \frac{2}{4} = \frac{3}{6}\]All of these fractions are equivalent because they simplify to the same fraction.

In the original exercise, we found that \(\frac{4}{5}\) is equivalent to \(\frac{20}{25}\). Even though the numbers are different, the relationship between the numerator and denominator is preserved. Understanding this helps in solving problems that ask you to find missing numerators or denominators.

Numerator

The numerator is the top number in a fraction. It shows how many parts of the whole we are considering. In the fraction \(\frac{4}{5}\), 4 is the numerator.

• It tells us about the parts we are focusing on
• Is affected by the multiplier when creating equivalent fractions

In the exercise, we needed to find the missing numerator. By understanding the multiplier that made the denominator 25, we also knew the numerator had to be multiplied by the same number, resulting in a new numerator of 20.

Denominator

The denominator is the bottom number of a fraction. It denotes the total number of equal parts that make up a whole. If the denominator in a fraction changes, it affects how the whole is divided.

• It indicates the size of each part of the whole
• It must be multiplied by the equivalent of the numerator's multiplier to keep the fraction unchanged

In our exercise, the denominator transitioned from 5 to 25. We made this change by using a multiplier, ensuring both the numerator and denominator changed by the same factor.

Multiplier

A multiplier is a number that you multiply both the numerator and denominator by to create an equivalent fraction. This ensures that the value of the fraction remains the same while the numbers change. The concept of a multiplier helps maintain the balance between the numerator and denominator.

• Essential for creating equivalent fractions
• Helps verify the relationships in fractions

In the exercise, we determined the multiplier to be 5 because multiplying the original denominator 5 by 5 equaled 25. Subsequently, the numerator was also multiplied by 5, leading to an equivalent numerator of 20.

Mathematics Problem Solving

Mathematics problem solving involves a series of logical steps to determine an answer. For fraction problems, this often includes identifying relationships and using multipliers.

• Understand the relationship between the numbers
• Use logical reasoning to find missing elements

In this exercise, the solution began by identifying how the fractions were related. By recognizing the increase in the denominator, we could determine the multiplier and apply it to both the numerator and the denominator. This kind of structured problem-solving is essential for improving mathematical thinking and accuracy.